1. Field of the Invention
The present invention relates to a method of seeking analytical solutions, and, more particularly, to a method of seeking semianalytical solutions to multispecies transport equations coupled with sequential first-order network reactions under conditions in which a groundwater flow velocity and a dispersion coefficient vary spatially and temporally and boundary conditions vary temporally.
In addition, the present invention relates to a method of seeking analytical solutions to complex problems such as inhomogeneous media and unsteady flow by combining a similarity transformation method of Clement with a generalized integral-transform technique (GITT).
2. Description of the Related Art
Typically, analytical solutions have been efficiently used to estimate and analyze groundwater contaminant transport.
Also, such analytical solutions may be very efficiently used to verify developed numerical solutions.
Further, thorough research into analytical solutions has been conventionally carried out, and conventional studies on analytical solutions include, for example, those disclosed in Domenico, 1987; Bear, 1979; Domenico and Robbins, 1985; Chilakapati and Yabusaki, 1999; Manoranjan and Stauffer, 1996.
However, such analytical solutions are limited only to analyzing single species contaminant transport, and the use thereof is even further restricted by complicated site situations.
For example, as in nuclear waste sites contaminated with radioactive materials and decayed daughter nuclides, and sites contaminated with chlorinated organic solvents such as PCE, TCE and biodegradable byproducts thereof, actual site situations are mainly associated with multispecies contaminant transport.
With reference to FIG. 1, a table in which conventional studies on analytical solutions are summarized is illustrated.
As illustrated in FIG. 1, Cho (1971) and Lunn et al. (1996) developed analytical solutions to three chemical species under simple boundary conditions coupled with sequential first-order reactions, van Genuchten (1985) developed analytical solutions to four chemical species coupled with sequential first-order reactions, and Sun et al. (1999a) developed analytical solutions to an arbitrary number of chemical species coupled with sequential first-order reactions.
Also, Sun et al. (1999b)developed analytical solutions to problems having sequential or parallel reaction networks, and Clement (2001) disclosed a similarity transformation method for solving multispecies transport equations coupled with any type of reaction.
As mentioned above, intensive research into analytical solutions to multispecies transport equations has been carried out, but the aforementioned conventional methods may be mainly applied only to multispecies transport equations under conditions of steady flow, uniform velocity and dispersion coefficient, and are developed only under fixed concentration boundary conditions, and thus limitations are imposed on such methods.
Actually, porous media are not for the most part homogeneous, and the media vary spatially and temporally all the time.
Hence, in the analysis of such complicated contaminant transport, it is not easy to apply typical methods such as Laplace or Fourier transform to multispecies transport equations, and it is also not easy to apply them to single species transport equations.
In regard thereto, Liu et al. (2000) proposed a generalized integral-transform technique (GITT) to semi-analytically solve one-dimensional transport-dispersion equations of single species contaminants in inhomogeneous media having spatially and temporally varying groundwater flow and dispersion coefficients.
More specifically, Liu et al. (2000) determined analytical solutions on the assumption that groundwater flow velocity, dispersion coefficient, and decay rate are arbitrary functions with respect to time and space and also that initial conditions and boundary conditions are arbitrary functions with respect to time and space.
However, GITT provides no procedures for determining semianalytical and analytical solutions to multispecies transport equations coupled with sequential first-order reaction networks in the inhomogeneous media and unsteady flow.
Accordingly, with the goal of solving the conventional problems as above, although it is preferred that there is provided a method of seeking semianalytical solutions to multispecies transport equations coupled with sequential first-order network reactions under conditions in which a groundwater flow velocity and a dispersion coefficient vary spatially and temporally, initial conditions vary spatially, and boundary conditions vary temporally, methods which satisfy such requirements have not yet been introduced.